TheHitchhikers GuideTo The Galaxy

Earth Edition

Units of Measurement

The 'root cause' of the loss of the spacecraft was the failed translation of English units into metric units in a segment of ground-based, navigation-related mission software. – Arthur Stephenson, NASA chairman of the Mars Climate Orbiter Mission Failure Investigation Board, 1999

Units are of vital importance in every civilisation as they are an essential means of communication. Extensive use of them is made in trade and science and they also play a significant role in everyday life. However, they are in no way an attainment of modern times. Whether a caveman wanted to pose with the size of the fish he had caught or a physicist wants to determine the relaxation time of an electron gas, units are needed everywhere on all levels of technological and cultural development.

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In order to have fun with them, however, a system of units must fulfil some basic requirements:

All people have to use the same units.

The set of units must be minimal.

The units must be defined as exactly as possible.

The units should be connected with each other in such a way that formulae using them become as simple as possible.

The Earth has followed a rocky road to such a system.

A Short History of Units

For the very first units, sizes of body parts were used: Ulna1(cubit), foot, thumb (inch). However people don't share the same body sizes and it was unbearable to have measured data depending on the measurer.

To overcome this, cultures began to keep standards of their units (standard lengths or bars, standard weights and so on) in temples or, later, in government buildings. The oldest known unit is the Babylonian foot from 2100 BC, which is of a length of 264.5mm. Political leaders often took such definitions very personally. For example, for several years the distance between the tip of the nose of King Henry I and his thumb was the defined British yard.

Under the rule of Charlemagne in the 9th Century AD, Europe had a uniform unit scheme, introduced in 789 AD. The subsequent disunity of Europe during the Middle Ages led to a wild assortment of completely unstandardised units, differing even from town to town. It is reported that in Baden, in southwestern Germany, in 1810 there were 112 different yards, 92 units of square measure, 65 units of volume, 183 units for cereals and 80 different pounds! This may have been fun if you wanted to cheat the farmer of your neighbouring village, however it made extra-territorial trade very difficult, and was useless for science.

But a solution had already been found, in Paris.

The SI

In 1791 Delambre and Méchain completed the measurement of the distance between Dunkerque (France) and Barcelona (Spain) which was used by a pan-European group of scientists to estimate the length of one Earth quadrant – one quarter of the circumference of the earth. Its 107th part was called one metre. Furthermore 1/1000 cubic metre was one litre and one litre of water had a mass of one kilogram. These new units were introduced in France in 1795 and in Germany in 1868. Seven years later, in 1875, 17 countries joined the metric union, called the CGPM (Conférence Général des Poids et Mesures).

In 1960 the CGPM decided to build a unit system, consisting of six (later seven) base units and their derivatives and called it Système International d'Unités: the SI. Since then, only the definitions of the base units have been improved, besides which tinkering the SI is state-of-the-art. It dominates the metric countries and international science. Its home is near Paris at the Bureau International des Poids et Mesures (BIPM).

So much for history.

Base Units of the SI

The SI developed from the metric system but today both terms can be used synonymously. It is worth mentioning that all metric countries have a different set of legal units, including some non-metric ones. The SI, in most cases, represents the core of the metric system, though.

The science that deals with measuring quantities and definition of units is metrology. Many countries run metrological institutes and partly they work as standardising authorities in their respective domains.

In regular international meetings these institutes decide about (re-)definitions of SI units. Since 1983 the seven base units have been defined as follows:

SI Base Units

Quantity

Name

Definition

Dimension

length

metre (m)

length of path travelled by light in vacuum during 1/299,792,458 of a second

L

time

second (s)

duration of 9,192,631,770 periods of radiation of the transition between the two hyperfine levels of the ground state of 133Cs

T

mass

kilogram (kg)

equal to the mass of the international platinum-iridium kilogram prototype kept under three glass bells in a safe (with three keys given to three different people) in the cellar of the Pavillion de Breteuil in Paris.

M

electric current

ampère (A)

constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 m apart in vacuum, would produce between these conductors a force of 2·10−7 N (newton) per metre of length

I

thermodynamic temperature

kelvin (K)

1/273.16 of the thermodynamic temperature of the triple point of water

K

amount of substance

mole (mol)

amount of substance which contains as many particles as there are atoms in 12 g (gram) of 12C

N

luminous intensity

candela (cd)

luminous intensity, in a given direction, of a source that emits monochromatic radiation of 540·1012 Hz (hertz) that has a radiant intensity in that direction of 1/683 W/sr (watt per steradian)

P

Derived Units of the SI

Another great convenience of the SI is that all units are either base units or simple products of them according to their dimension – no coefficients, no zero point shifts. This is called a coherent system. Therefore, it's superfluous to list any conversion factors: there aren't any.

SI Derived Units

Quantity

Dimension

Name

frequency

T−1

hertz (Hz)

force

LMT−2

newton (N)

pressure

L−1MT-2

pascal (Pa)

energy

L2MT−2

joule (J)

power

L2MT−3

watt (W)

Photometric Units

luminous flux

P

lumen (lm)

illuminance

L−2P

lux (lx)

Radioactivity

activity

T−1

becquerel (Bq)

dose equivalent

L2T−2

sievert (Sv)

absorbed dose

L2T−2

gray (Gy)

Electrical Units

charge

TI

coulomb (C)

potential difference

L2MT−3I−1

volt (V)

resistance

L2MT−3I−2

ohm

conductance

L−2M−1T3I2

siemens (S)

capacitance

L−2M−1T4I2

farad (F)

magnetic flux

L2MT−2I−1

weber (Wb)

magnetic flux density

MT−2I−1

tesla (T)

inductance

L2MT−2I−2

henry (H)

Others

catalytic activity

T−1N

katal (kat)

refractive power

L−1

diopter (dpt)

The disadvantage of a coherent system is that some units are tiny; the pascal for example, while others are huge like the tesla. Therefore there are a few units in use that are very closely related to SI units – namely by powers of ten. These are:

SI Related Units

Name

Value

Use

barn (b)

10−28 m2

cross section in atomic and nuclear physics

gauss (G)

10−4 T

magnetic flux density

bar

105 Pa

pressure

litre (l)

10−3 m3

volume (mostly of liquids)

tonne (t)

1000 kg

weight

ångström (Å)

10−10 m

dimensions in crystallography

micron (µ)

10−6 m

microscopic lengths

fermi (fm)

10−15 m

lengths on sub-atomic scale

Metric Prefixes

A much more general solution for titchy or gigantic units are prefixes. Placing them directly before units will change their values by powers of ten. For example, 1 nF (one nano farad) is equal to 10−9 F = 0.000 000 001 F.

Although there are no further rules for using prefixes, there is a sort of conventional etiquette:

Yotta, zetta, zepto and yocto are not regarded as part of the SI.

Use deci only with metre.

Use centi only with litre and metre.

Use hecto only with litre and pascal.

Don't use deka.

Metric Prefixes

Factor

Name

Symbol

Factor

Name

Symbol

(1024

yotta

Y)

10−1

deci

d

(1021

zetta

Z)

10−2

centi

c

1018

exa

E

10−3

milli

m

1015

peta

P

10−6

mikro

µ

1012

tera

T

10−9

nano

n

109

giga

G

10−12

pico

p

106

mega

M

10−15

femto

f

103

kilo

k

10−18

atto

a

102

hecto

h

(10−21

zepto

z)

101

deka

da

(10−24

yocto

y)

In information theory the same prefixes are applied to its fundamental unit, the byte (B). However, in a majority of cases of use, 1 MB actually represents 1024·1024 B = 1,048,576 bytes, rather than a strictly accurate 1,000,000 B. Therefore some metric purists called for clear distinction. The result was published in 1998. Now 1024 B are one kibibyte (1 KiB), 10242 B are one mebibyte (1 MiB) and 10243 B are one gibibyte (1 GiB). This well-meant, albeit slightly academic approach, while approved by many standardisation authorities, hasn't yet made it into everyday practice.

Minutes and Hours

Minutes and hours as measures of time are not part of the core of the SI but they are very closely linked with it, particularly in everyday use of the SI system. 60 seconds are one minute (min) and 60 minutes are one hour (h).

Most people think that 24 hours are one day, which is – again, strictly speaking – only an approximation. The length of the day is adjusted to the course of the sun, which is a little bit irregular. The same is true for the week and even more for months and years. The realm of actual standard units of measurement ends at the hour.

Non-SI Units in Use

Maybe life would be too boring if all people used SI units. Sometimes non-SI units are used because they make a certain specific calculations easier, however in most cases people are simply not bold enough to break with tradition, or are reluctant to 'learn' the SI unit.

For a typical example of this type of usage, a car with 150 hp has a good deal of power – but a car with 112 kW? Well, as you might guess, they are exactly the same.

Even scientists frequently prefer non-SI units. Tradition plays a role here, too, but sometimes they have a real excuse: natural units. These are useless in practice but have the convenient effect that natural constants become equal to one, which keeps formulae and calculations simple. For example, Einstein's famous equation E=mc2 is reduced to a very elegant E=m.

In the following tables numbers in bold face are exact per definitionem.

United Kingdom

To facilitate being part of the European Union, the UK was urged to abandon all imperial units. The pint as a measure of beer or milk is still legal, as is buying pizza by the inch, but you can hire the other units for a decent fine of £5000.

Practically, however, the imperial units are still widely used. A big supermarket chain tested metric units for six months, then switched back to pounds and ounces because the customers had felt 'puzzled and bemused'.

Imperial Units

Name

Value

Remarks

Length

nautical mile (n mile)

1.853184 km

ocean navigation

mile

1.609344 km

80 ch = 1760 yd

yard (yd)

91.44 cm

36 in

foot (ft)

30.48 cm

12 in

inch (in, '')

2.54 cm

milliinch (mil, ''')

25.4 µm

1/1000 in

hand

10.16 cm

4 in; height of horses

furlong (fur)

201.168 m

220 yd; horse racing

chain (ch)

20.1168 m

22 yd

fathom

1.8288 m

2 yd; depth of sea

Area

acre

4046.86 m2

4840 yd2 = 10 ch2

Volume

gallon (UKgal)

4.54609 l

pint (pt)

0.568261 l

1/8 UKgal

fluid ounce (fl oz)

28.41304 ml

1/160 UKgal

barrel

163.659 l

36 UKgal; for beer

Mass (avoirdupois system)

pound (lb)

0.45359237 kg

ton

1.016047 t

2240 lb

stone

6.3502932 kg

14 lb; weight of people

ounce (oz)

28.34952 g

1/16 lb

dram (dr)

1.771845 g

1/16 oz

Mass of gems and precious metals

troy ounce (oz tr)

31.1034768 g

pennyweight (dwt)

1.55517384 g

1/20 oz tr

Others

poundal (pdl)

0.138255 N

1 lb·ft/s2; force

horsepower (hp)

745.700 W

power

British thermal unit (Btu)

1.05506 kJ

energy

therm

105.506 MJ

105 Btu; energy

USA

The USA are bottom of the metric league. Metric units have been legal national standard in the US since 1866, though nobody cares. Curiously enough, metric units are used in most US science fiction series. Hopefully they are not complete fiction. The National Institute for Standards and Technology (NIST) tries hard to enforce the use of the SI in the USA but has hitherto had little success.

Germany

Since the old days in Baden the measurement situation has been greatly improved. As in most continental countries, the use of non-SI units has become very rare here. Responsible is the PTB in Braunschweig, which calls itself the 'Guardian of Units'.

Peculiar Units in Use

Temperature

Temperature is basically a rather simple concept. There is a point of absolute zero temperature (0 K – ie, no heat at all) and from there you can measure temperature just like length or mass. Unfortunately this leads to very odd values for everyday temperatures – around 300 K. Therefore people use other scales with other zero points eg, °C, °F. This makes it necessary to give formulae rather than mere factors to convert between temperature units.

Rankine and reaumur are obsolete temperature units. The Rankine was intended to be for Fahrenheit what kelvin is for Celsius – a thermodynamic temperature scale. Celsius made the race all over the world, however, which meant the end of the rankine.

Fahrenheit (1714) said 'temperature of a mixture of water, ice and ammoniac'.

Reaumur (1730) said 'temperature of a mixture of water and ice'.

Celsius (1742) said 'temperature of boiling water'. Later people found this potty and changed it to Reaumur's definition.

Because all of this was arbitrary, these units were disfigured with degree signs. The real zero point remained a mystery until W Thompson, or Lord Kelvin, determined it in 1848. Hence the kelvin has no '°' and is allowed to wear metric prefixes like mK or µK.

Pureness of Precious Metals

It is not only that outside the SI there are many units for the same quantity, sometimes even within the SI one unit can indicate multiple quantities. The carat, for example, is a unit of mass, but it can also be used to indicate how much gold, silver etc is contained in a given alloy. 25/6 · carat gives the percentage of pureness. Thus a say 22-carat gold chain contains 92% gold. Pure gold has 24 carats, so higher values would be a metallurgical miracle (or simply a lie).

Logarithmic Units

A simplified conception of the Weber-Fechner law says that our perception of external stimulation is proportional to the logarithm of that stimulation. This is only a very rough approximation, but it explains the need for logarithmic units, especially if our senses are involved.

Mathematical note: Unfortunately the logarithm can't stand units. In order to get rid of them one has to divide the quantity by another quantity of the same unit. This second quantity is part of the definition and is usually very small to avoid negative values.

decibel (dB): Actually the decibel is a far more general measurement, but mostly it is used to indicate the loudness of sound. Let p be the sound pressure, then 10·log(p/20µPa) is the loudness in decibels. 40 dB is normal small talk, 140 dB is a jet engine and 180 dB is lethal. However our ears are not equally sensitive for all frequencies; the maximum is at approx. 1000 Hz. The phon scale takes this into account (at 1000 Hz, phon and dB are identical).

magnitudines (m): Astronomers specify the brightness of stars in magnitudines (or magnitudes) which is based on an ancient Greek system. In this the brightest stars had 1m and the faintest 6m. Today's exact definition is m=−2.5·log(s/s0). Here s indicates the illuminance of the star seen from earth and s0 is adjusted in such a way that the Pole Star has exactly 2.12m. Later it was found that the brightness of the Pole Star is in fact slightly variable, but astronomers are very patient people.

Richter scale: It's a measure for the intensity of earthquakes. Let A be the maximal amplitude of the ground oscillation 100 km away from the epicentre of an earthquake, then log(A/1µm) is its value on the Richter scale. Earthquakes beyond 6 are regarded as really big ones, for them A can be several metres.

pH scale: A measure of acidity. Let n be the molar concentration of H+-ions in a solution, then −log(n/(1mol/l)) is its pH value. In water, values from 0 (strong acid) up to 14 (strong alkali) are possible. Pure water has a pH of 7 (neutral), yet drinking water is a little bit lower.

bit: The bit can be seen as a logarithmic measure of information. If a certain data container can represent N different possible contents (eg numbers) it has (3.322·log N) bit. For example, a variable that can have values from 0 up to 255 is a 3.322·log 256 = 8 bit variable. 8 bit form one byte. There are even larger units like word, double word and paragraph, however their sizes are not clearly standardised. See also the discussion of binary metric prefixes above4.

Angular Units

The only mathematically legitimate unit of plane angle is the radian (rad). It's the length of the corresponding arc in the unit circle. For all non-mathematicians: 2π rad is the full angle, π/2 rad5 is a right angle.

For solid angle the steradian (sr) is used. It's the area of the corresponding sphere segment of the unit sphere. The full sphere has 4π sr.

There aren't actually any other sensible – or equally intuitive – possibilities to measure angles, but mankind is not very sensible and highly imaginative. One degree (°) is (π/180) rad (so 360° are the full angle). Every degree is divided into 60 arc minutes (') end every arc minute into 60 arc seconds ('').

One gon is (π/200) rad. Gon is also called grade or new degree (g). Every new degree is divided into 100 new minutes (c)6 and every new minute into 100 new seconds (cc). Remarkably enough the gon is a legal unit, however almost nobody uses it.

There are other angular units which are even more bizarre. All the units in the following table, for example – legal or not – must also be regarded as obsolete units, excluding degree and its derivatives.

Angular Units

Name

Value

Full Angle

Remarks

Plane angle

degree (°)

1.745329·10−2 rad

360°

π/180 rad

(arc) minute (')

2.908882·10−4 rad

21600'

(1/60)°

(arc) second ('')

4.848137·10−6 rad

1,296,000''

(1/3600)°

(arc) gon (gon), grade (grade), new degree (g)

1.570796·10−2 rad

400 gon

(1/200) rad

new minute (c), cgon

1.570796·10−4 rad

40000c

(1/100) gon

new second (cc)

1.570796·10−6 rad

4,000,000cc

(1/10000) gon

revolution (r)

6.283185 rad

1 r

2π rad

mil (mil), (artilleristic) point (¯)

9.817477·10−4 rad

6400 mil

π/3200 rad

(nautical) point ('')

0.1963495 rad

32 point

π/16 rad; ocean navigation

Solid angle

square degree ((°)2)

3.046174·10−4 sr

41252.97(°)2

(π/180)2 sr

Obsolete Units

The following passages contain units not used anymore. However you may still come across them sometimes.

It is virtually impossible to list all units, there being almost as many potential units as there are human beings. If you encounter an unknown unit and want to look for its value, a rather good starting point is one of the national genealogy institutes, especially if it is a historical unit. Otherwise the metrological authorities are worth a try.

Obsolete Scientific and International Units

Name

Value

Dimension

Use

x-unit (KX)

1.00202·10−10 m

L

spectroscopy

didot point (dd)

0.376 mm

L

continental typography

cicero (cc)

4.531 mm

L

continental typography

pica

4.218 mm

L

typography

geographical mile

7.421591 km

L

eötvös (E)

10−9 s−2

T−2

acceleration in geophysics

franklin (Fr)

3.33564·10−10 C

TI

electric charge

debye (D)

3.33564·10−30 Cm

LTI

electric dipole momentum

clausius (Cl)

4.1868 J/K

ML2T−2K−1

1 cal/K; entropy

rutherford (Rd)

106 Bq

T−1

radioactive activity

jansky (Jy)

10−26 J/m2

MT−2

irradiation

biot (Bi)

10 A

I

electric current

maxwell (M, Mx)

10−8 Wb

L2MT−2I−1

magnetic flux

gilbert (Gb)

0.795775 A

I

10/4π A; magnetic tension

new candle (NK)

1 cd

P

luminous intensity

international candle (IK)

1.019 cd

P

luminous intensity

nit (nt)

1 cd/m2

PL−2

luminous density

stilb (sb)

104 cd/m2

PL−2

luminous density

apostilb (asb)

0.318310 cd/m2

PL−2

(1/π) cd/m2; luminous density

lambert (La)

3183.10 cd/m2

PL−2

(1/π) cd/cm2; luminous density

phot (ph)

104 lx

PL−2

illuminance

litre atmosphere (l atm)

101.325 J

ML2T−2

energy

gamma (γ)

10−9 kg

M

mass

pond (p)

9.80665·10−2 N

MLT−2

force

dyne (dyn)

10−5 N

MLT−2

force

poise (P)

0.1 Pa s

ML−1T−1

dynamic viscosity

stokes (St)

10−4 m2/s

L2T−1

kinematic viscosity

erg

10−7 J

ML2T−2

energy

enzyme unit (U)

1.6667·10−8 kat

NT−1

1 µmol/minute

United Kingdom

Obsolete Imperial Units

Name

Value

Remarks

Length

rod, perch, pole (rod)

5.0292 m

5.5 yd

Area

rood

1011.71 m2

40 rod2 = 1/4 acre

circular inch

5.06707 cm2

(π/4) in2

Volume

peck

9.09218 l

2 UKgal

bushel

36.3687 l

8 UKgal

quarter

290.9498 l

64 UKgal

chaldron

1.309273 m3

4 quarter

quart (qt)

1.136523 l

1/4 UKgal

gill

142.0652 ml

1/32 UKgal

minim (min)

0.0591939 ml

1/180 fl oz

fluid drachm (fl dr)

3.55163 ml

60 min

fluid scruple

1.183877 ml

20 min

hogshead (hhg)

238.4809 l

only for liquids

Mass

short ton

0.907185 t

2000 lb

hundredweight (cwt)

50.8024 kg

8 stone = 112 lb

cenral (sh cwt)

45.3592 kg

100 lb

quarter

12.7005864 kg

1/4 cwt

grain (gr)

64.7989 mg

1/7000 lb

slug

14.59390 kg

Mass of drugs

apothecaries' ounce (oz ap)

31.1034768 g

480 gr

drachm

3.8879346 g

60 gr

scruple

1.2959782 g

20 gr

Germany

The following units are all Prussian apart from one. Prussia was the north-eastern part of Germany, and the industrial and scientific heartland of the country. The exception is the Deutsche Meile, a general, historical German unit.

Obsolete German Units

Name

Quantity

Value

Deutsche Meile (mile)

length

7.5 km

Linie (line)

length

0.218 cm

Zoll (inch)

length

12 Linien = 2.615 cm

Fuß (foot)

length

12 Zoll = 31.39 cm

Elle (cubit)

length

25.5 Zoll = 66.69 cm

Klafter

length

6 Fu = 1.883 m

Rute (rod)

length

2 Klafter = 3.766 m

Meile (mile)

length

2000 Ruten = 7.532 km

Schritt (yard)

length

1/10000 Meile = 75.32 cm

Russia

Today Russia uses the metric system in almost all measurements – even in their airplanes, which is potentially dangerous since western control towers normally use imperial units. Note that the Russian dujm and fut are exactly equal to the inch and foot. The names are mere transcriptions from Cyrillic.

Status and Future of the SI

Today, 94.5% of the world's population uses the metric system. On the whole only three countries haven't included it into their industrial standards: The USA, Myanmar and Liberia.

Most other systems have been connected to the SI. Since 1959, the inch is defined using the metre, and the avoirdupois, troy and apothecaries' systems via the kilogram.

The Convention of the Metre of 1875 is still the basis of all international agreement on units of measurement. There are now 48 member countries in the BIPM, including all major industrialised nations. However, practical enforcement of usage varies heavily from country to country.

The US National Institute of Standards and Technology (NIST) has launched a 'Metric Program' under the banner 'Toward a Metric America'. Hopefully they will succeed in converting the US to usage of metric units. However, doing this comprehensively and quickly would probably cost hundreds of billions of dollars.

Fortunately the German Physikalisch-Technische Bundesanstalt (PTB) needn't do this. They focus on more accurate definitions of units. For example the kilogram is defined in a rather anachronistic way by a prototype that is likely to change over the centuries. Their project 'Avogadro' ia an attempt at an overhaul.

1The thin long bone in the human forearm.2For food mostly kcal = 1000 cal are used. People often refer to it as plain calories.3Note that 'counterpart' does not imply equal value.4By the way 3.322 is an approximation for 1/log 2.5You can omit the 'rad', although it is considerably less confusing for non-mathematicians to include it.6You may also call this a centigon.7Note that other sources may differ in spelling.

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